### Video Transcript

Simplify three ๐ฅ๐ฆ to the power of three over negative ๐ฅ squared ๐ฆ to the power of
three, where ๐ฅ๐ฆ is not equal to zero, and evaluate this expression when ๐ฅ equals
negative one and ๐ฆ equals negative two.

Now the first stage of this problem is to actually simplify three ๐ฅ๐ฆ all cubed over
negative ๐ฅ squared ๐ฆ all cubed. Well, first of all, what weโre gonna do is actually multiply out the parentheses. So weโre gonna start with the numerator. And what we can actually think is if we have three ๐ฅ๐ฆ all cubed, and that actually
means each of these terms cubed, so itโs like three cubed multiplied by ๐ฅ cubed
multiplied by ๐ฆ cubed, which is gonna give us 27๐ฅ cubed ๐ฆ cubed. Thatโs cause three cubed, three multiplied by three is nine, multiplied by another
three is 27.

And then weโve got ๐ฅ cubed ๐ฆ cubed. Well, then weโre gonna have the same for the denominator. So weโve got negative ๐ฅ squared all cubed multiplied by ๐ฆ cubed. Well, therefore, what weโre gonna use is actually one of our index laws to rewrite
this because weโve got an index law that tells us that if we have ๐ฅ to the power of
๐ then to the power of ๐, this is equal to ๐ฅ power of ๐ multiplied by ๐.

So therefore, our denominator is gonna be negative ๐ฅ to the power of six, because we
had ๐ฅ power of two to the power of three, multiply three and two, we get six, then
๐ฆ cubed. So now to actually simplify this, what weโre gonna do is apply another index law,
which tells us that if we have ๐ฅ to the power of ๐ divided by ๐ฅ to the power of
๐, then this is gonna be equal to ๐ฅ to the power of ๐ minus ๐.

So therefore, our fully simplified answer is gonna be negative 27๐ฅ power of negative
three. Thatโs because if we have ๐ฅ to the power of three divided by ๐ฅ to the power of six,
weโll actually subtract them from each other. So we have three minus six, which gives us negative three. And then if you have ๐ฆ to the power of three divided by ๐ฆ to the power of three, we
subtract three from three, which gives us zero. So weโre left with negative 27๐ฅ to the power negative three.

Okay, so great! Thatโs the first part of the question answered. But now what we need to do is actually evaluate the expression when ๐ฅ is equal to
negative one and ๐ฆ is equal to negative two. So therefore, to actually evaluate this expression and find out how much it is when
๐ฅ is equal to negative one and ๐ฆ is equal to negative two, all we have to do is
actually substitute in ๐ฅ is equal to negative one. And this is because when we simplified it, we actually donโt have any ๐ฆ terms
left.

So therefore, we can actually rewrite this as negative 27 over negative one all
cubed. And thatโs cause we substituted in ๐ฅ as negative one. And the reason we can write it like this is because weโve got another relationship we
know, which is that ๐ฅ to the power of negative ๐ is equal to one over ๐ฅ to the
power of ๐. So this is gonna give us negative 27 divided by negative one. And thatโs because negative one cubed is negative one. So therefore, as a negative divided by a negative is positive, it just gives us an
answer of 27.

So now even though weโve actually fully simplified and found the answer, what weโre
gonna do is actually check because what we can do is check by substituting in ๐ฅ
equals negative one and ๐ฆ equals negative two into the original expression. And we can actually see there if weโve actually managed to fully simplify correctly
and find the correct value, because this should give us the same value.

So therefore, if we substitute in our values for ๐ฅ and ๐ฆ, weโll get three
multiplied by negative one multiplied by negative two all cubed over negative then
negative one squared multiplied by negative two, and then this is all cubed. So therefore, this is gonna give us six cubed. Thatโs because we had three multiplied by negative one, which is negative three,
multiplied by negative two, which is gonna give us six, cause a negative multiplied
by a negative is positive, so six cubed, over, then weโve got two cubed. Thatโs because we had negative negative one squared. Well, negative one squared is positive one. But we make that negative cause the negative sign in front, so weโve got negative one
multiplied by negative two, which is positive two. And then this is all cubed, which is gonna give us 216 over eight, which gives us 27,
which is the answer we got when we worked out the first method.

So great! What we can say is that, fully simplified, three ๐ฅ๐ฆ to the power of three over
negative ๐ฅ squared ๐ฆ to the power of three is negative 27๐ฅ power of negative
three. And the value of this expression when ๐ฅ is equal to negative one and ๐ฆ is equal to
negative two is 27.